Predicting Survival Probability of NASA Aircraft Engines

Cox Proportional Hazards (CPH) Model

What is it?

Limitations

Mathematical Formulas

• Survival Function: \(S(x) = \int_{x}^{\infty} f(x)dx\)

• Hazard Function: \(h(x) = \lim\limits_{\Delta x \rightarrow 0} \frac{P[x \leq X < x + \Delta x|X \geq x]}{\Delta x}\)

• CPH Model Hazard Function: \(h(t|\mathbf Z) = h_0(t)\text{exp}(\sum\limits_{k=1}^{p} \beta_kZ_k)\)

• Proportional Hazards Ratio: \(\frac{h(t|\mathbf Z)}{h(t|\mathbf Z*)} = \text{exp}[\sum\limits_{k=1}^{p} \beta_k(Z_k - Z_k^*)]\)

• Cumulative Hazard Function: \(H(x) = \int_0^x h(u) du\)

• Concordance Index: \(C = \frac{c + \frac{t_x}{2}}{c + d + t_x}\)

• Survival Probability: \(S(t) = e^{-H(t)}\), where \(H(t)\) is the above cumulative hazard function

Assumptions

There are four assumptions for CPH:

Visualization

The two common model visualizations are:

Evaluation and Survival Probability

Data Structure

The data selected for this project comes from a study on propagation modeling that NASA completed, specifically focusing on the engine two testing and training datasets (Saxena et al. 2008).

• Each engine in the NASA data has an unknown amount of wear, manufacturing variation, and sensor noise

• There are three operation setting fields and twenty-one sensor measurement fields

• A column indicating status was added to both the testing and training datasets with 0 indicating the machine has not failed and 1 indicating the machine has failed

• The data was then merged together to make one grouped dataset

Exploration

• 519 engines in the combined data

  • 260 engines in training data

  • 259 engines in testing data

Visualization

Creating CPH Model

The table provides the model number, the covariates used, the AIC and BIC from the stepwise regression if applicable, and the concordance index.

Model for the continuing analysis will have all covariates except sm16 and sm19.

CPH Model Forest Plot

Checking Assumptions

Model Results

Conclusion